Animating the development of an organic structure
Q: How to morph mesh shapes with different amount of vertices?

A: There is not an efficient way in Blender to animate mesh with varying topology (from what I know).

Blender is able to generate this kind of data as export of fluid sim or metaballs into Alembic (.abc) format. This format can be also imported back into Blender. A bit of setup can be found under Modifier tab - Mesh Sequence Cache (see Manual). But in general - you can transform only on object level (Loc/Rot/Scale or use modifiers on top of this format).

Like a hack of your proposal I tried to animate sculpted object with shapekey. When topology become to its limit I duplicate the object and continue to sculpting with dyntopo on specific part. It worked in kind of, but its was so painfull with pour result.



Q: How to animate growing an organic structure

A: You can try - Particles with Metaballs growing by gradient texture.
In this sample I used just a tree structure (addon Sapling), but you can model gland by your self (or some cool script probably exists for this kind of structure).


add MetaBall object (for later use)
create some gland structure (edges only)  
subdivide edges evenly into dense vertex mesh


Particle System  


add Particle System to this object with option Render as Object
as object select MetaBall
set a number of particles the same as amount of gland vertices
(it can be found in edit mode in bottom row)
you would have to set particles Velocity to zero and disable Gravity under Scene properties to stop particles to move


Animation


animation is done by illusion - particles scaled in time
particle scale is controlled by Particles > Texture
add new Texture and go to Texture properties
set Blend > Spherical, Influence > Size > enable, Mapping > Object
(You can animate size directly in this panel, but its more flexible use another object. I added Empty into scene.)
additionaly you can enable Colors > Color Ramp to adjust appearence of size in time, so it will looks like growing thickness


It has its limits to animate by texture, but you can probably get some more variations by adding another Particle Texture to "break" spherical growing shape (I didnt try that).